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I want to prove that $$∂U = \overline U \cap \overline{X\setminus U}.$$

Not too sure where to start with this question so any help would be appreciated.

I am also struggling to prove $U^{\circ} \subset U$ for any $U$. I have the following definitions: $$∂U = \overline U \setminus U^{\circ},\ U^{\circ} = X\setminus \overline{(X\setminus U)}.$$

Any help is much appreciated.

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  • $\begingroup$ Just to confirm $\ U^{\circ} = X\setminus \overline{(X\setminus U)}$ is the given definition of interior, right? $\endgroup$ – ir7 Mar 1 '18 at 15:20
  • $\begingroup$ @ir7 yes sorry that is the definition of interior $\endgroup$ – Nicole Mar 1 '18 at 16:36
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With the given definition of interior, we have:

$$∂U = \overline U \setminus U^{\circ} =\overline U \setminus( X\setminus \overline{(X\setminus U)}) = \overline U \cap \overline{(X\setminus U)}.$$

Also:

$$ U^{\circ} =X\setminus \overline{(X\setminus U)} \subset X\setminus (X\setminus U) = U, $$ as $X\setminus U \subset \overline{(X\setminus U)}$.

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  • $\begingroup$ thank you so much! :) $\endgroup$ – Nicole Mar 1 '18 at 17:42
  • $\begingroup$ You’re welcome. $\endgroup$ – ir7 Mar 1 '18 at 17:44

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